So I asked a similar question to this on Math Stack Exchange a couple of weeks ago, did a bounty, and I didn't receive any answers.

I am struggling a bit with a part of my research (on CS).

Suppose I have a finite set $X$, and two functions $f,g:X \rightarrow \mathbb{R}_{\geq 0}$. These functions are particularly unique, meaning that $\forall x,y \in X, f(x)\neq f(y) \Rightarrow |f(x)-f(y)|>\epsilon$ for some fixed $\epsilon$. In other words, If the functions are not equal then they are at least apart a distance of $\epsilon$.

Now here is where I am interested, Let's say I fix a line $y_{x_1}(x)=f(x_1)+g(x_1)x$ and a point $C^*$ on the x-axis. Now I consider a second line $y_{x_2}(x)=f(x_2)+g(x_2)x$ and vary $x_2$ over all the elements in $X-\{x : f(x)=f(x_1),g(x)=g(x_1) \}$. Let us say that each of these lines, $y_{x_i}$ intersect $y_{x_1}$ at point $(x_{1i},y_{1i} )$. Now I am interested to find the minimum possible distance from $C^*$ to any of the $x_{1i}$.

Now if this problem was for simply any functions $f,g$ with the unique condition dropped, then the problem would trivially reduce to $0$ since I can find a line $y_{x_i}$ that intersect $y_{x_1}$ at exactly $C^*$ by varying $f(x_1)$ and $g(x_1)$ slightly. However, the functions in place here have the annoying condition that you have to change $f$ and $g$ with at least a perturbation of size $\epsilon$. This $\epsilon$ might not be enough to make the lines intersect.

So My question is, is there any sort of "Calculus" per say, that handles with such functions which are almost continuous, but can't be changed except by at least some small perturbation?

So I asked a similar question to this on Math Stack Exchange a couple of weeks ago, did a bounty, and I didn't receive any answers.

I am struggling a bit with a part of my research (on CS).

Suppose I have a finite set $X$, and two functions $f,g:X \rightarrow \mathbb{R}_{\geq 0}$. These functions are particularly unique, meaning that $\forall x,y \in X, f(x)\neq f(y) \Rightarrow |f(x)-f(y)|>\epsilon$ for some fixed $\epsilon$. In other words, If the functions are not equal then they are at least apart a distance of $\epsilon$.

Now here is where I am interested, Let's say I fix a line $y_{x_1}(x)=f(x_1)+g(x_1)x$ and a point $C^*$ on the x-axis. Now I consider a second line $y_{x_2}(x)=f(x_2)+g(x_2)x$ and vary $x_2$ over all the elements in $X-\{x : f(x)=f(x_1),g(x)=g(x_1) \}$. Let us say that each of these lines, $y_{x_i}$ intersect $y_{x_1}$ at point $(x_{1i},y_{1i} )$. Now I am interested to find the minimum possible distance from $C^*$ to any of the $x_{1i}$.

Now if this problem was for simply any functions $f,g$ with the unique condition dropped, then the problem would trivially reduce to $0$ since I can find a line $y_{x_i}$ that intersect $y_{x_1}$ at exactly $C^*$ by varying $f(x_1)$ and $g(x_1)$ slightly. However, the functions in place here have the annoying condition that you have to change $f$ and $g$ with at least a perturbation of size $\epsilon$. This $\epsilon$ might not be enough to make the lines intersect at $C^*$.

So My question is, is there any sort of "Calculus" per say, that handles with such functions which are almost continuous, but can't be changed except by at least some small perturbation?